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Biomedical Signal and Image Processing
(a)
(b)
(c)
(d)
FIGURE 3.18 (a) Original image, (b) filtered image with A = 1.1, (c) filtered image with
A = 1.15, and (d) filtered image with A = 1.2.
3.3.3.3 Derivative Filters
As we saw in the previous sections, image blurring can be caused by averaging.
Since averaging is simply the discrete version of spatial integration, one can expect
that spatial differentiation would result in image sharpening. This observation about
spatial differentiation is the main idea behind a family of sharpening filters called
“derivative filters.” In order to find suitable masks for spatial differentiation, we need
to study the concept of differentiation in digital 2-D spaces more closely.
Since an image is a 2-D signal, instead of simple 1-D differentiation, the directional differentiations must be calculated in both horizontal and vertical directions.
This leads to spatial gradient defined as follows:
⎡ ∂f ⎤
⎢ ⎥
∂x
∇ = ⎢ ⎥
(3.7)
f ⎢ ⎥
∂f
⎢ ⎥
⎣ ∂y ⎦
Biomedical Signal and Image Processing
(a)
(b)
(c)
(d)
FIGURE 3.18 (a) Original image, (b) filtered image with A = 1.1, (c) filtered image with
A = 1.15, and (d) filtered image with A = 1.2.
3.3.3.3 Derivative Filters
As we saw in the previous sections, image blurring can be caused by averaging.
Since averaging is simply the discrete version of spatial integration, one can expect
that spatial differentiation would result in image sharpening. This observation about
spatial differentiation is the main idea behind a family of sharpening filters called
“derivative filters.” In order to find suitable masks for spatial differentiation, we need
to study the concept of differentiation in digital 2-D spaces more closely.
Since an image is a 2-D signal, instead of simple 1-D differentiation, the directional differentiations must be calculated in both horizontal and vertical directions.
This leads to spatial gradient defined as follows:
⎡ ∂f ⎤
⎢ ⎥
∂x
∇ = ⎢ ⎥
(3.7)
f ⎢ ⎥
∂f
⎢ ⎥
⎣ ∂y ⎦